Notable Properties of Specific Numbers

The number of ways to arrange a
2×2×2 Rubik's Cube. As there are no centre
cubelets to determine the orientation, one corner is considered to have
a fixed, defined orientation. The other 7 can be put into any of the
7!=5040 possible positions, and all but one can be rotated
into any of 3 different rotations (the total rotation of all 8 pieces
always adds up to 360o).

According to early Hindu mythology, the mahayuga or "great age" is
a period of time consisting of four consecutive ages, lasting 1728000,
1296000, 864000 and 432000 years for a total of 4320000. They placed
themselves and all of humanity in the fourth of these ages, see
432000. The great age repeats many times; the longer
periods in the Hindu cosmological calendar are described under
622080000000000. See also 8640000000.

This is the "original" Smith number, and was in fact the
telephone number of someone named Smith. A Smith number is a number
for which the sum of the digits is equal to the sum of the digits of
its prime factors: 4937775 = 3×5×5×65837, and 4+9+3+7+7+7+5 =
3+5+5+6+5+8+3+7. Numberphile has a video on it:
4937775 - Smith numbers. See also 22 and
1×1010694985.

The length (in metres) of the major (transverse) axis of the
ellipsoid (or oblate spheroid) used by the WGS 84 model to
approximate the shape of the Earth. This is very close to the average
equatorial radius of the Earth, if you measure based on where the
gravitational field is equal to that at sea level. (The sea, being a
fluid, tends to equalise its height profile such that gravity is the
same at all points on its surface, and the WGS 84 model is calibrated
to agree with sea level as closely as possible). See the Geoid
article for an explanation of how the geoid (the "gravitational
equipotential surface") differs from the actual surface of the Earth.
Apart from following the sea height as just mentioned, it tends to be
underground below any significantly elevated land. Local changes of
density in the mantle and crust add lots of variation.

If the earth were a sphere and the meter agreed exactly
with its original definition, this would be exactly 20 million divided
by pi.

This number is an exceptional counterexample to the
abc conjecture. The abc conjecture states that, given two
relatively prime numbers a and b, the sum of the distinct prime
factors of a, b and of their sum c=a+b, called rad(abc),
is "almost always" bigger than c. For example when a=7 and
b=33=27, c=34=2×17, which makes
rad(abc)=2×3×7×17=714, quite a bit bigger than c.
6436343 is special because it is so far in the other direction:
a=310×109, b=2, c=235=6436343, and
rad(abc)=2×3×23×109=15042, much less than c.

When written in the UK style (horizontal strokes through the 7's,
and a straight, not curved, vertical stroke for the 9) and held up to
a mirror, this number looks fairly like "PISS OFF". See also
176, 22024, and 5318008

8114118 is a palindrome, and the 8114118thprime
143787341 is also a palindrome. This is the smallest such number,
after a few early trivial cases (like 11 which is the 5th prime).
The prime is a member of A46941 and its index is in
A46942. It was discovered by Carlos Rivera35, and is
followed by 535252535.

This is both a prime and a palindrome, the next-larger palindrome
prime is 9136319. This would not be very special if it were not also
for the fact that, in the digits of π, the digits 9136319
appear starting at position 9128219.

The first of a set of 5 consecutive primes that are spaced
an equal distance apart: 9843019, 9843049, 9843079, 9843109 and
9843139 are all prime, there are no primes in between, and the spacing
between each one and the next is 30. 9843019 is the lowest
number with this property; the next is 37772429. See also 47,
251, 121174811 and
19252884016114523644357039386451.

107 appears in the definition of the
vacuum permeability constant μ0, also called the
"permeability of free space", in the curious formula:

μ0 = 4π/107 N/A2

where N is newtons and A is Amperes. Those
are both long-established units in the SI system, so one might wonder
where this 107 comes from.

A current formal definition of the ampere is "the constant current
which will produce an attractive force of 2×10-7 newtons per
metre of length between two straight, parallel conductors of infinite
length and negligible circular cross section placed one metre apart in
a vacuum". There is that factor of 107 again, right in the
definition of the ampere. Persuing this further goes right back to the
definition of μ0, a circular definition.

To reveal the origin of the 107 we have to look at the history of
the ampere unit and the discovery of the force between two electric
wires carrying current, a phenomenon first demonstrated by
Andre-Marie Ampere in 1820. The (historical) original definition
of the modern unit is 1/10 of the unit now called an abampere,
which in turn was "the amount of current which generates a force of
two dynes per centimetre of length between two wires one centimetre
apart". A dyne is a g cm/s2, and a newton is
a kg m/s2, so a dyne is 10-5 newtons. In the units that were
common in Ampère's time, μ0 was simply 4π:

So we see that the 107 in the modern definition of μ0 is a
relic of the old centimetre gram second system of units.
Converting from dynes to newtons diminished the
value by 105; measuring the force per meter of wire rather than per
centimeter increased the value by 102; moving the wires from a
distance of 1 cm to 1 meter canceled that 102 out, and measuring
the current in amperes rather than abamperes reduced the force by
102 because Ampère's force is proportional to the product of the
currents in the two wires and both measurements change by a factor of
10 (the number of amperes in an abampere).

107=10000000 is a unit of the (Asian) Indian number name system.
It is called crore when needed (primarily in Indian dialect of
written English). In Iranian usage a crore is 500000. See
also 10000 and 100000.

A product of two non-overlapping sets of consecutive integers:
17297280 =
8×9×10×11×12×13×14 =
23×32×2×5×11×22×3×13×2×7 =
2×3×7×26×5×13×6×11 =
63×64×65×66. This type of
match is is more "unlikely" than that demonstrated by
19958400 because it requires more prime factors to work
out right after rearranging. See also 720,
175560, and Sequence A064224.

Combined fuel economy of a Toyota Prius, in SI units (50 miles per
gallon converted to meters (of distance traveled) per cubic meter
(volume of fuel consumed)). See xkcd 687 and
3.1418708596056; see also 137.035.

19958400 = 3 × 4 × (5×6×7×8×9×10×11) =
(5×6×7×8×9×10×11) × 12 = 12! / 24. This is the product of
the integers 3 through 11, and also the product of integers 5 through
12. There are an infinite number of ways to construct a number with
this sort of pattern, all of which have a similar form: two
consecutive numbers at the beginning (in this example 3×4) get
replaced by their product, an oblong number (in this example
12), at the end. The general form is:

The length (in meters) of the IUGC standard meridian.
This represents the length of a line from one pole of the Earth to the
other (crossing the equator midway, i.e. at about the 10,002-kilometer
point). It is an international standard agreement, and is a sort of
average of meridians at different longitudes75. The original
definition of meter was based on the meridian and would have had this
number be exactly 20000000. The original determination of the meter's
length, based a massive seven-year surveying project, established a
meridian length that was too small.

Later improvements in understanding about the Earth's shape and
extensive established use of the meter for non-surveying purposes made
it necessary for the unit to diverge from its original meridian-based
definition. The total change in length of the meter through this
process was about 195 parts per million. The meter ended up being a
bit "shorter", and the initial meridian measurement was too short (by
a greater amount), so the average meridian is now known to be nearly
20,004 km. See also 1852.

The number of seconds in common (non-leap) year:
365×86400. Although Leap seconds are called
"intercalary", they are effectively part of the year because the leap
second occurs during the day, local time (for example, in a time zone
7 hours away from UTC, the clocks would go from "16:59:59" to
"16:59:60" to "17:00:00") so a common year with a leap second would be
31536001 seconds long.

The number of SI seconds in a tropical year, according to
xkcd 1061. The "Earth Standard Time" system,
which is "simple, clearly defined, and unambiguous", defines a year by
the following rules:

1 year = 12 months; 1 month = 30 days; 1 day = 1440 minutes (= 24
hours 4 minutes); 1 minute = 60 SI seconds. This gives 1 year =
31190400 SI seconds.
For 4 hours every full moon, run clocks backward. Full moons
happen every synodic month, which is 29.530588853×86400.001 =
2551442.9... SI seconds. After going backwards for 4 hours, the clocks
have to go 4 hours forward before continuing, so 8 hours = 8×3600 = 28800
seconds will be added. This increases the average length of a year by
a ratio of (2551442.9...+28800)/2551442.9... = 1.01128773..., making it
31542468.8... SI seconds.
The non-prime-numbered minutes of the first full non-reversed
hour after a solstice or equinox happen twice. The 17th prime number
is 59, so there are 43 non-prime minutes in an hour. There are two solstices
and two equinoxes per year, so this rule adds 43×60×4 = 10320 seconds
to the year, for a year of 31552788.8... SI seconds.

An approximation of the number of SI seconds in a mean
tropical year, as experienced during the years
1995-2012. This is based on the average rate of rotation of the Earth
during that period (see mean solar day) combined with
the tropical year length for the year 2000 (which is
in mean solar days). The figure for the length of the
mean solar day has less precision because of the
variations in Earth rotation rate on short timescales49,125,
due to weather and ocean currents, etc. whereas the year length figure
represents an average over a period of several years.

In words, the number of SI seconds in the mean tropical year
multiplied by the Sun's mean rate of motion in arc-seconds per century
is equal to the number of seconds in a Julian century times the number
of arc-seconds in a full circle.

This is the number of seconds per year according to the
Gregorian calendar (averaged over a 400-year period):
365.2425 times 86400. It is an exact integer
but is just an average; the number of seconds in any particular year
is always either 31536000 or 31622400.

Randall Munroe[216] found the approximation
754=31640625, which is a better approximation than the
popular (among physicists) π×107 =
31415926.535... .

The number of seconds in a leap year: 366×86400.
Although Leap seconds are called "intercalary", they are
effectively part of the year because the leap second occurs during the
day, local time (for example, in a time zone 7 hours away from UTC,
the clocks would go from "16:59:59" to "16:59:60" to "17:00:00") so a
leap year with a leap second would be 31622401 seconds long.

This is the number of different ways that one can visit the state
capitols of the 48 contiguous states in the United States, passing
through each state only once. The same route in reverse does not count
as a distinct route, and one end of the trip must be in Maine because
it only borders one other state. The answer, and a description of
algorithms used to calculate it, are in Knuth [165] section
7.1.4 (Binary Decision Diagrams), (p. 255 in the 2011 edition).

Type this on a calculator and read the display upside-down; it (sort
of) says "SHELL OIL". In the 1970's there were a bunch of joke "word
problems" that instructed the reader to enter some sort of formula
(example: 30 × 773 × 613 - 1 = × 5 =) to produce an answer
that is read as a word by holding the calculator upside-down. For this
purpose the digits 0,1,2,3,4,5,7,8,9 were used to represent O, I, Z,
E, H, S, L, B and G respectively, so the answer/punchline could be any
word or phrase using only these letters. See also 31337 and
5318008.

The "Tyson Code", better known as 0073735963 or 007-373-5963, a
cheat code in the Nintendo videogame Mike Tyson's Punch-Out that
takes the player directly to the final match against Tyson himself.
See also 573, 9001, and 1597463007.

A myriad myriad, and the largest number mentioned in the
Bible (Hebrew תנ"ך (Tanakh) or Christian Old
Testament): Daniel 7:10, "...
and ten thousand times ten thousand stood before him, ..." (King
James version). It is probably not a coincidence that 108 was also
the largest number for which the Greeks had a name; the book of Daniel
reached its final form well after Alexander conquered the entire
Levant region. See also 666.

108 is 億 in China (yì, dàng) and Japan (oku),
where they construct numerals on the basis of 10,
100, 10000, 108, and higher powers of 104.
This system closely resembles the Knuth -yllion
naming system for very large powers of 10. (See also
my list of large numbers in Japanese)

The first of a set of 6 consecutive primes that are spaced an equal
distance apart: 121174811, 121174841, 121174871, 121174901, 121174931
and 121174961 are all prime, there are no primes in between, and the
spacing between each one and the next is 30. 121174811 is the
lowest number with this property; it was first discovered in 1967 by
L. J. Lander & T. R. Parkin. Along with 2, 3, 251 and
9843019, forms a sequence (Sloane's
A6560) that is thought to be infinite, but it is
very hard to discover the next one. No one has yet discovered the
first set of 7 consecutive primes; such a set would have to have a
spacing of 210 or a multiple of 210; see
19252884016114523644357039386451. See also 47,
251 and 9843019.

Newcomb's coefficient giving the average rate of motion of the Sun
across the sky (or equivalently, the rate of Earth's motion in its
orbit, relative to the stars) in units of arc-seconds per century. One
might think this number should just be 129600000, but
the Earth's axial precession and other effects prevent this.

This number, 227 or 233, is equal to this rather
memorable sum of cubes:
5003+2003+1003+603+123. Another
way to express this fact is:

ln((5322)3 + (5223)3 + (5222)3
+ (3×4×5)3 + (3+4+5)3) = ln(2) 33

Scary but true: I actually discovered and verified this property of
227 by doing the math in my head. I already knew most of the powers
of 2 up to 224=16777216. And, like tens of other kids around the
world, I learned the squares up to 202 and the cubes up
to 123 in grade school. One day I decided to double 224 a few
times to get 227, then noticed the 217728, which
looks a lot like 216 and 1728 stuck together. It
was then fairly easy to see the rest, since 134 is 125 plus
8 plus 1. See also 2097152.

This is a 9-digit number containing each of the digits 1 through 9,
and equal to the sum
96+89+73+62+57+44+31+25+18, in which each
of the digits occurs exactly once as a base and exactly once as an
exponent. Inder J. Tenaja calls numbers of this type "flexible power
selfie numbers", and found a total of 25 of them (with 389645271
being the largest).

In early 2009, one David Horvitz (an artist who enjoys posting
unusual ideas on his blog) suggested that people should take a photo
of themselves standing in front of a fridge or freezer with the door
open and their head in the freezer, then share it online (e.g. with
Instagram or Flickr) tagged with the number 241543903. The idea caught
on (becoming an internet meme) and an image search for this number
will now return dozens of such photos.

The smallest 9-digit number that, when written in three rows of 3
(as in one block of a Sudoku puzzle) forms a 3×3
magic square. There are 7 others: 294753618,
438951276, 492357816, 618753294, 672159834, 816357492, and 834159672.

299792458 is the speed of light in meters per second. In 1983 by
international agreement, the meter was redefined in terms of the speed
of light, and as a result the constant for the speed of light is now
exactly 299792458 meters per second. The second, in turn, is defined
as precisely 9192631770 times the frequency of photons in
a Caesium maser-based atomic clock. See also 2.54,
8.987552×1016, 1.6160×10-35 and
5.390×10-44.

The speed of light was first calculated from astronomical
measurements in 1710 by Ole Romer, but had to be expressed as a
ratio to the speed of Earth in its orbit (or equivalently, in terms of
certain unknown Solar System distances and known light travel times)
because the size of the astronomical unit had not yet been
determined to sufficient accuracy; this would not come until the late
1700's (see 149597870691 for more).

A meter is also just about equal to the length of a pendulum with a
period of precisely two seconds (a seconds pendulum, the length is
close to 994 millimeters). In fact, this definition was proposed as
the standard unit of length over 100 years before the original
Metric system became official, and for most of the 18th century
it was one of two competing proposals. The other proposal (based on
the size of the Earth) was chosen because the period of a pendulum
depends on where it is measured. (See 20003931.4585 for
more about the meridian measurement and its errors).

It is a strange coincidence that the gravitational acceleration at
Earth's surface (9.8 meters per second2) times the length of
Earth's year (about 31557600 seconds) is about 310000000
meters per second, just a little bit bigger than the speed of light.
There is no significance to this coincidence, it's just kind of cool.
See also 3.14187.

The mean acceleration due to gravity on the Earth's surface, times
the number of seconds in a
mean tropical year. This happens to be only a few
percent larger than the speed of light. This serves
as a guideline to some basic limits on long-duration manned space
flight. Since astronauts would probably need to experience no more
than about 1.1 or 1.2 times normal gravity during their trip, it would
take a few years (even from the astronaut's own relativistic frame of
reference) to make the trip even to the nearest stars.

This 9-digit number contains one each of the digits 1 through 9, and
has the additional property that the first two digits (38) are a
multiple of 2, the first 3 digits (381) are a multiple of 3, and so on
up to the whole thing being a multiple of 9. You can see a bit of
symmetry in the digits: the first three digits (381) plus the last 3
(729) add up to 10×111, and the middle 3 (654) plus itself in
reverse (456) also adds up to 10×111. This type of number is called
polydivisible, and this one is also pandigital in that it contains
each digit (except 0) exactly once. There are lots of such numbers if
you don't care about having one each of the digits 1 through 9. See
also 3816547290, 30000600003, and
3608528850368400786036725.

456790123 has the "370-property": it is equal to the
average of all possible permutations of its digits. Since there are 9
digits, there are 9! = 362880 permutations. That would
take a really long time to add up to take an average, but we can save
a lot of work by noting that each digit occurs in each position an
equal number of times. For example, the digit "4" will appear in each
position in exactly 1/9 of the permutations. This effectively means
that we can compute the average much more quickly just by using one
representative permutation with each digit in each possible position.
In this case, that can be done by computing:

where the 9 terms are the original number rotated into all possible
positions (like the multiples of 142857). If you take
this sum (on a 10-digit calculator) you'll find that the average is
equal to the original number, 456790123. These numbers of this type
(first pointed out to me by reader Claudio Meller) are discussed
more fully on their own page.

535252535 is a palindrome, and the 535252535thprime
11853735811 is also a palindrome. This is similar to
8114118 and was discovered by Giovanni Resta. The prime
is a member of A46941 and its index is in A4694235

This is a power of 2, and a 9-digit number in which all 9 digits are
different. There is no 10-digit power of an integer in which each of
the digits 0 through 9 appears once. See also
295147905179352825856.

The (false) Polya conjecture stated that
positive integers with an odd number of prime factors always outnumber
those with an even number of prime factors. In this case, the "number
of prime factors" is sequence A001222, in which the same prime
can be counted twice (so for example 8=23, 12=22×3 and
30=2×3×5 are all counted as having 3 prime factors). But
the conjecture turns out to be false in a small region starting at
906150257 and extending up to 906488079.

The number of seconds from the 1st January 1970 until the 1st
January 2001. This is 11323 days, i.e.
(365×31+8)×86400 seconds, because 2001 is 31
years after 1970 and there were 8 leap years during that period. The
number appears as an offset in time/date calculations when converting
between the UNIX epoch and the epoch used in the MacOS Cocoa framework
("Core Foundation"). Both use 00:00:00 GMT as the moment the counting
starts, and ignore leap seconds. Cocoa defines the constant
kCFAbsoluteTimeIntervalSince1970 equal to 978307200.0L

This is (1000-1)3 = 10003-3×10002+3×1000-1, and its
reciprocal 1/997002999 =
0.000000 001 003 006 010 015 021 028 036 045 055... gives us
the triangular numbers. This happens because the
generating function of that sequence is 1/(x-1)3. For more on
this, see my separate article
Fractions with Special Digit Sequences; see also
89, 99.9998, 199, 998,
9801, and 9899.

A billion in the "short scale" system
used in the United States, and comparatively recently adopted by the
UK and other English-speaking countries. Most other countries use the
"long scale" in which a "billion" is 1012.

This difference in usage (109 versus 1012) came into being at
a time when it didn't matter to most people. But thanks to many
factors (population growth, inflation, prosperity, technology, and
education) numbers in the billions are now very common in the news and
in everyday speech. The honor associated with the name millionaire
in the early 1900's now belongs to the billionaire. We often hear of
costs and deficits in the billions; many of our computers have
billions of bytes of storage capacity and perform billions of
operations per second.

109 is an estimate of the processing power (in floating-point
operations per second) embodied in a human retina. The retinas perform
image processing to detect such things as edge movement and boundary
direction. The figure is based on a resolution of roughly 106
pixels, a speed of 10 changes per second, and 100 FLOPs per pixel. See
also 1018.

As you can see, there are two different sets of patterns. As long as
n is a multiple of an odd number, 10n+1 fits at least one of
the patterns. The numbers excluded by this are of the form
102i+1: 11, 101, 10001, 100000001, 10000000000000001, etc.
(Sloane's A80176, the "base 10 Fermat numbers").
There is no easy factorisation pattern for them. ([151] pp. 137-138)

This is the second example in a series of near-misses to Fermat's last
theorem discovered by Ramanujan, of which 1729 is the famous
first example. 1030301000 is 10103, and is just 1 greater than the
sum of 7913 and 8123. See this article and the
336365328016955757248 entry for details.

This is the decimal value of the hexadecimal integer constant
0x5f3759df that comprises the central mystery to the following bit of
code, which is mildly famous among bit-bummers and purports
to compute the function f(x) = 1/√x:

This code actually works. It performs four floating-point multiplys,
one floating-point add, an integer shift, an integer subtract, and two
register moves (FP to Int and Int back to FP). It generates the
correct answer for the function to within three decimal places for all
valid (non-negative) inputs except infinity and
denormals.

The hex value 0x5f3759df is best understood as an IEEE
floating-point number, in binary it is
0.10111110.01101110101100111011111. The exponent is 101111102,
which is 190 in decimal, representing 2(190-127) which is 263.
The mantissa (after adding the hidden or implied leading 1 bit) is
1.011011101011001110111112, which is 1.43243014812469482421875 in
decimal. So the magic constant 0x5f3759df is
1.43243014812469482421875×263, which works out to the integer
13211836172961054720, or about 1.3211...×1019. This
is (to a first-order approximation) close to the square root of
2127, which is about 1.3043...×1019. The reason
that is significant is that exponents in 32-bit IEEE representation
are "excess-127". This, combined with the fact that the
"exponent.mantissa" floating-point representation crudely approximates
a fixed-point representation of the logarithm of the number (with an
added offset), means that you can approximate multiplication and
division just by adding and subtracting the integer form of
floating-point numbers, and take a square root by dividing by two
(which is just a right-shift). This only works when the sign is 0 (i.e.
for positive floating-point values).

Here are some example values of numbers from 1.0 to 4.0 in IEEE
single-precision:

Here I have shown the sign, exponent and mantissa separated by dots.
Since the logarithm of 1 is zero, the value for 1.0
(0.01111111.00000000000000000000000) can be treated as the "offset".
If you subtract this offset you get these values, which approximate
the logarithm of each number:

From this it is easy to see how a right-shift of the value for 4
yields the value for 2, which is exactly the square root of 4, and a
right shift of the value for 2 gives the value for 1.5, which is a bit
higher than the square root of 2. Over a full range of
input values, the right-shift and addition of the magic constant gives
a "piecewise linear" approximation of 1/√x.

The constant "0x5f3759df" is most commonly cited as being found in the
Qrsqrt function of "game/code/qmath.c" in the source code of thevideogame Quake III. It is attributed to John Carmack, but the same
hack appears in several earlier sources going as far back as 1974
PDP-11 UNIX.

David Eberly wrote a paper[175] describing how and why the
approximation works.

Chris Lomont[178] followed up with investigation into its
origins, getting as far as a claimed credit to Gary Tarolli of Nvidia.
He thoroughly analyzes the piecewise linear approximation for odd and
even exponents and proposes 0x5f375a86 as being slightly
better, and a similar constant 0x5fe6ec85e7de30da for use with 64-bit
IEEE double precision.

David Eberly then wrote a longer explanation[205] analyzing
the constant 0x5f3759df along with some other candidates (like
0x5f375a86 and 0x5f37642f). It describes efforts to
discover why and how this value originally got chosen; with
inconclusive results.

An earlier example of code calculating the square root in this way
(approximation via a single shift, possibly with an add or subtract,
no conditional testing, and Newton iteration) was described by Jim
Blinn in 1997, where we find the following code: (see [163]).

This is actually pretty weird. We are shifting the floating-point
parameter  exponent and fraction  right one bit. The low-order bit
of the exponent shifts into the high-order bit of the fraction.
But it works.
- Jim Blinn ([163] page 83)

The same article discusses several similar functions including ones
that include one iteration of Newton's method. Here are his
inverse square root functions:

which is effectively performing an integer right-shift on the 16 high
bits of the input value, then adding a constant similar to the
constants in the above examples, and putting the result back into a
floating-point register before proceeding with the Newton's method
calculations. Only the upper part of the mantissa is being shifted,
but that's good enough. A man page from Feb 1973 (Third Edition UNIX)
suggests that the routine existed as early as then.

The number of seconds from the 1st January 1904 until the 1st
January 1970. This is 24107 days, i.e.
(365×66+17)×86400 seconds, because 1970 is 66
years after 1904 and there were 17 leap years during that period
(including 1904 itself). It is the offset between the UNIX epoch and
the epoch used in the old "Classic" MacOS; see 978307200
and 3061152000 for more.

The number of seconds from the 1st January 1904 until the 1st
January 2001. This is 35430 days, i.e.
(365×97+25)×86400 seconds, because 2001 is 97
years after 1904 and there were 25 leap years during that period
(including 1904 itself). The number appears as an offset in time/date
calculations when converting between the UNIX epoch and the epoch used
in the old "Classic" MacOS. Both use 00:00:00 GMT as the moment the
counting starts, and ignore leap seconds. The Cocoa / Core Foundation
framework defines the constant kCFAbsoluteTimeIntervalSince1904equal to 3061152000.0L

In late 2014 a Twitter friend and I undertook a
challenge to
find the smallest (integer, not starting with any 0's) number that
does not appear in any Google search results (or, at the very least,
try to estimate
how many digits it would have).

Using Fermi Estimation (see Randall Munroe's
what-if 84), I estimated that: there
are 1010 people, each has 1 webpage, each with 1000 words; but only
1% of these are devoted to long lists of unique numbers (like invoice
numbers, telephone numbers, etc.), and probably 90% of them are either
small and duplicate each other somewhat, or are big and leave gaps.
Answer: the smallest integer not indexed by Google is
probably 10 digits long.

He and I spent a while trying numbers, and pretty quickly found that
the 10-digit numbers seem to be almost all taken. 11-digit examples
were easy to find. After just 10 minutes or so we had gotten down to
the very low 11 digits (my best was
10826746091,
his was
11170063270).

He kept looking for 10-digit numbers, and noticed that there seem to
be extensive lists of primes, but not of composites. He discovered
that
6255626957 =
109×3803×15091 was unknown to Google, and soon after found that
the Marshall Islands have country code +625. (The islands have 7-digit
phone numbers but only enough people to use a small fraction of them,
thus offering a possible explanation). Shortly after this, he and
another had found
3112066128 =
24×3×64834711. (Internationally, +31 is
The Netherlands but 9 digits
must be added; within the U.S. 311 is an N11 code; so there are no
10-digit telephone numbers starting with 311).

Clearly this number will be indexed soon after appearing on this
page, so I would call it a "likely upper bound" for whatever number is
actually the smallest positive integer not in any Google result.
Within a few years, perhaps all 10-digit numbers will have appeared
somewhere.

3432948736 is the smallest number N such that N = 2N mod
10K, where K=10. In other words, 2 to the power of 3432948736
ends in the digits 3432948736. This is a member of a sequence
(Sloane's A121319) that is thought to be endless.
It has the nice property that each member of the sequence adds a digit
to the previous one. For example, 28736 ends in 8736, 248736
ends in 48736, 2948736 ends in 948736, and so on.

The only 10-digit pandigital polydivisible number in base 10: For
each n from 1 to 10, the first n digits of this number, taken as
an n-digit number, are divisible by n. For example, the first 3
digits are 381, and 381 is divisible by 3. The whole thing is
divisible by 10 since it ends in 0, and any permutation of the 10
digits would be divisible by 9 since the sum the 9 digits is
45 which is a multiple of 9. But the other divisibility
requirements impose tight constraints. See 381654729 for
more about the pattern in these digits. See also
6210001000, 30000600003,
3608528850368400786036725, and
101.845773452536×1025.

The Human population of the Earth according to the
Arecibo message, which was transmitted in 1974. A more modern
estimate is 6771000000. This is possibly the most
dangerous number anyone has ever sent in any communication, because as
Cassiday notes77, "Aliens who correctly interpret this will
know how large an army to send".

The "self-describing number" described by Numberphile's
James Grime in the video
Maths Puzzle: The self descriptive number. It is
the unique ten-digit number in which the first digit (6) tells how
many zeros the number has; the second digit (2) tells how many 1's,
etc., viz.:

One might think that searching for such a number would require
checking all 9,000,000,000 ten-digit numbers; but that's not needed
because the digits must sum up to 10. As James mentions in the
solution video, the search can be reduced even
further by realising any solution must be one of the
partitions of 10, of which there are
only 42.

Because 13 is 2×7-1, 13! is the magic constant for this
"multiplicative" 7×7 magic square:

27

50

66

84

13

2

32

24

52

3

40

54

70

11

56

9

20

44

36

65

6

55

72

91

1

16

36

30

4

24

45

60

77

12

26

10

22

48

39

5

48

63

78

7

8

18

40

33

60

which is built on the principle of doing an elementwise multiplication
(Hadamard product) on the following
two components:

3

5

6

7

1

2

4

9

10

11

12

13

1

8

2

4

3

5

6

7

1

12

13

1

8

9

10

11

7

1

2

4

3

5

6

8

9

10

11

12

13

1

5

6

7

1

2

4

3

o

11

12

13

1

8

9

10

4

3

5

6

7

1

2

1

8

9

10

11

12

13

1

2

4

3

5

6

7

10

11

12

13

1

8

9

6

7

1

2

4

3

5

13

1

8

9

10

11

12

both of which satisfy the row, column, and diagonal requirements, but
with repeated numbers. Is is quite efficient, in the sense that it
uses 53.8% of the numbers from 1 to 7×13=91, or 67% of
those that remain after casting out all primes greater than 13.

This is 29 primorial,
2×3×5×7×11×13×17×19×23×29 and has a really
easy-to-remember digit pattern: 646 969 323 0. The pattern
results from the properties of 1001=7×11×13 and
2001=3×667=3×23×29, which multiplied together give
2003001, and 323=17×19.

This is the answer to a puzzle that appeared on billboards in 2004.
The billboards stated:

{first 10 digit prime in consecutive digits of e} . com

This little bit of nerd sniping led the solver
to another, harder puzzle also involving digits of e. That puzzle,
if solved, brought the user to a website soliciting resumes, potentially
resulting in a call from someone at Google.

This is the first prime number in alphabeticalorder in the
English language: "eight billion eighteen million eighteen thousand
eight hundred and fifty-one". It was found by Donald Knuth. All other
numbers that occur earlier in alphabetical order (like 8 and
8018018881) are composite. ([151] p. 15 footnote)

Neil Copeland has suggested32 that 8000000081 is the
alphabetically first prime, based on the spelling "eight billion and
eighty-one". The use of and is common outside the U.S. (I have
confirmed reports from the UK and New Zealand). Knuth, consistent with
his statement in [146], does not use and.

This is "Coulomb's constant", also called the "electric force
constant" or "electrostatic constant", and is c2/107 N/A2
where c is the speed of light in metres per
second, N and A are the units newton and ampere.
Since c is defined to be precisely 299792458, Coulomb's_constant is
precisely 8987551787.3681764 N m2/(A2s2); the units are
equivalent to metres per farad.

Frequency (in Hz) of microwave radiation used as the basis of the
Caesium-133 atomic clock. This number is part of the official
definition of the second (the basic unit of time).

The length of the second is originally derived from the rotation of
the Earth and time-division decisions by the Babylonians, among other
things (see 86400). Also, the rotation rate of the Earth
keeps changing  it has changed by 19 parts per billion in the past
100 years49, enough to mean that this number could have been anywhere
from about 9192631680 to about 9192631860 (and the number defining the
meter and the speed of light, see 299792458, could
have been anywhere from 299792455 to 299792461).

The largest number that can be formed from the digits 1, 2 and 3
using the ordinary functions addition, multiplication and/or
exponents. It slightly edges out 231=2147483648 because
log(3)/log(2) is greater than 31/21. The next number in this sequence
is 101.0979×1019.